Models Separating the Internal and External Filtration Processes

For convenience in the modeling, Pang and Sharma (1994) divide the entire filtration process into two phases:

1 the initial internal cake filtration, and

2 the later external cake filtration.

They separate these two filtration phases by a "transition time" after which the particle migration into porous formation becomes negligible and an external filter cake begins forming over the injection well formation face. Sharma and Pang (1997) assumed that transition from the internal to the external cake filtration occurs when the porosity, ɸ, of the formation face decreases to a minimum critical value, ɸ, by particle deposition, below which particle invasion into porous media is not possible. Their models apply for single phase water flow in the near wellbore region. Hence, the effect of the oil-water two-phase flow during the initial water injection period is neglected because this initial period is relatively short.

Transition Time

where b is an empirical constant and X,0 is the filtration coefficient without particle deposition. Although more sophisticated expressions are available in the literature (See Chapter 8), they used Eq. 19-15 for simplicity. Thus, invoking Eq. 19-15 into Eq. 19-14 yields the following expression, similar to the rate equation for particle deposition in pluggable pathways given by Gruesbeck and Collins (1982):

At the transition time, the porosity attains the minimum critical porosity, ɸ , and the maximum critical volume fraction of the deposited particles
becomes a* = ɸ0 - ɸ* . Under these conditions, Eqs. 19-17 and 18 can be used to obtain the following expressions, respectively, for the transition time:

Internal Filtration Models

Here, v represents the interstitial velocity of the fluid phase and n denotes the fraction of the pore space occupied by the particle deposits in porous
media. X' is the product of the deposition rate constant, K, and the interstitial velocity of the flowing suspension. In the following, their formulations are presented in a manner consistent with the rest of the presentation of this chapter. The damage of a core plug by the injection of a dilute particulate suspension can be described by means of the volumetric balance equations of the suspended and deposited particles in porous media, given, respectively, by (Wennberg and Sharma, 1997):

The analytical solution of Eqs. 19-24 through 27 used by Pang and Sharma (1994) implies some simplifications. It applies for the injection of dilute suspension of particles. Therefore, the effect of small amount of particle deposition, a, compared to the initial porosity can be neglected, the deposition rate coefficient is assumed constant, and a constant rate injection is considered. Thus, the analytic solution for constant ɸ~ɸ0, X,~X0 , and u = u0 can be adopted from Rhee et al. (1986) as:

where the term inside the square brackets expresses that cf is a function of (t-ɸx/u). Considering that cf(t) = cf is constant and c0(x) = Q and a0(x) = 0 in the laboratory core flow tests, Pang and Sharma (1994) simplify Eqs. 19-29 and 30, respectively, as:

Pang and Sharma (1995) assumed that the permeability reduction primarily occurs by pore throat plugging. Therefore, they estimated the permeability of the porous formation as a harmonic average permeability of the combined plugged and unplugged regions as:

where fp represents the volume fraction of the deposited particles contributing to pore throat plugging, Kp denotes the permeability of the plugged region near the pore throats, and Km is the permeability of the formation matrix, assumed to remain constant, which is equal to the initial nondamaged permeability, K0 (i.e., Km =K0}. Therefore, Pang and Sharma (1995) rearranged Eq. 19-35 for the relative or fractional retained permeability of the porous formation undergoing particle deposition from dilute suspensions as, inferred by Payatakes et al. (1974):

Pang and Sharma (1994) simplify Eq. 19-41 by considering that the injection front reaches the outlet end of the core rapidly. Therefore, neglecting the damage during the short period of time until the front reaches the core outlet, Eqs. 19-39 and 41 yield for xf = L the following equation indicating that the reciprocal injectivity ratio is a linear function of time:

External Filtration Models

Considering the formation of an incompressible external cake without any particle invasion into the core plug, Pang and Sharma (1994) expressed the harmonic average permeability of the cake and the core system as:

Thus, substituting Eq. 19-48 into 45 and considering that the filter cake thickness is much smaller than the length of the core plug (i.e., hc « L),
they obtained the following expression indicating that the reciprocal injectivity ratio is a linear function of time:

in which x, v, z, and p are some empirical parameters and aM is the maximum of the volume fraction of the deposited particles necessary to make the filtration coefficient of porous media zero. This equation indicates that the filtration coefficient is equal to one when there is no deposited particles in porous media, and the filtration coefficient becomes zero when the volume fraction of deposited particles reaches a certain characteristic value of maximum aM. Chiang and Tien (1985) developed an empirical correlation as:

where ɸ is the porosity in fraction. D is the coefficient of diffusion for the Brownian motion of particles. Wennberg and Sharma (1997) analyzed the measurements of the filtration coefficient reported by various investigators and determined that these data mostly indicate power law-type relationships to the volumetric flux, the suspended particle size, and the porous media grain size as:

Diagnostic-Type Curves for Water Injectivity Tests

Pang and Sharma (1994, 1997) identified four distinct type curves that can be used for interpretation of the water-quality tests. They justified these type curves with experimental data obtained from the literature as shown in Figure 19-3. Type curve 1 is a straight line indicating the formation of an incompressible external filter cake or a thin internal cake near the injection face of the core plug according to Eq. 19-49. The slope remains constant. Type curve 2 is for the similar case, but applies for compressible cakes. In this case, the porosity and permeability of the cake decrease by increasing filtration pressures. As a result, the slope of the curve increases with the filtration time or pore volume injected. Type curve 3 refers to a deep particle invasion and pore filling in the core plug, leading to a slower gradual permeability decrease. As a result, the slope of the curve decreases with the filtration time. Type curve 4 may be an S-shaped or other types of curves indicating a shift of the dominance of the different damage mechanisms during the filtration process.

Todd, A. C., et al., "The Application of Depth of Formation Damage Measurements in Predicting Water Injectivity Decline," SPE Paper 12498, presented at the Formation Damage Control Symposium held in Bakersfield, California, February 13-14, 1984.

Todd, A. C., et al., "The Value and Analysis of Core-Based Water Quality Experiments as Related to Water Injection Schemes," SPE Paper 17148, presented at the SPE Formation Damage Control Symposium held in Bakersfield, California, February 8-9, 1988.

Wennberg, K. E., & Sharma, M. M., "Determination of the Filtration Coefficient and the Transition Time for Water Injection Wells," SPE Paper 38181, Proceedings of the 1997 SPE European Formation Damage Conference held in the Hague, The Netherlands, June 2-3, 1997, pp. 353-364.